Effect of Hunting on Red Deer

Modelling Fecal Cortisol Metabolites

Nikolai German, Thomas Witzani, Ziqi Xu, Zhengchen Yuan, Baisu Zhou

Dr. Nicolas Ferry - Bavarian National Forest Park / Daniel Schlichting - StabLab

31 Jan 2025

Agenda

The Background

The Data

The Models

The Wrap-up

Motivation

Hunting activities might induce stress for red deer, even if non-lethal.
Goal: analyze how the distance in space and time to the (last) hunting event affects the stress in deers.

FCM Level and Gut Retention Time

Faecal Cortisol Metabolites (FCM) are substances collected from feces of animals.
FCM level [ng/g] is a measure of stress level. Higher Stress \(\implies\) higher FCM level.
Stress \(\Rightarrow\) secretion of certain hormones \(\Rightarrow\) gut retention \(\Rightarrow\) FCM.
FCM level does not represent stress level when defecating.
Gut retention time \(\approx\) 19 hours.

The Approach

Model FCM levels on spatial and temporal distance to hunting activities
Expectation: FCM levels higher when closer in time and space

Agenda

The Background

The Data

The Models

The Wrap-up

The Data-Generating Process

A deer roams freely in the Bavarian Forest National Park.
Its movement is tracked by a GPS collar.
A hunting event happens.
After some time, the deer defecates. This is a defecation event.
After some time, researchers go to the defecation location and collect an FCM sample.

The Datasets

Movement Data
Hunting Events
FCM Data
Reproduction Success

The Movement Data

Contains the location of the 40 collared deer
Period: Feb 2020 - Feb 2023
Tracking: Movement is tracked in hourly intervals

The Hunting Events

Contains location and date of hunting events
Observations: 1270 events
890 Observations with complete timestamp

The FCM Data

Contains information of 809 faecal samples, including

the FCM level [ng/g],
the time and location of sampling,
to which deer the sample belongs,
when defecation happened.

The Reproduction Success

Contains information about of 16 collared deer about

if they were pregnant in one year,
if they were accompanied by a calf in one year.

Distance Approximation via Interpolation

TBD: Illustration.

Relevant Hunting Events

We introduce 4 parameters:

Gut retention time (GRT) low [hours],
Gut retention time (GRT) high [hours],
Distance threshold [km],
Proximity criterion (“last”, “nearest”, or “highest score”)

A hunting event is considered relevant to a FCM sample, if

the time difference between experiencing stress (hunting) and defecation is between the GRT thresholds, and
the distance between the deer and the hunting event is \(\leq\) distance threshold.

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by the proximity criterion:

the closest in time (“last”),
the closest in space (“nearest”), or
the one with the “highest score.”

The scoring function is defined as TBD.

Illustration

GRT lower = 14 hours
GRT upper = 50 hours
Distance threshold = 20 km

The Fused Data

Extract Temporal Features

The Fused Data

Interpolate Movements

The Fused Data

Compute Spacial Distances

The Fused Data

Add Pregnancy Data

The Fused Data

Identify Events

The Fused Data

Finish Datasets

We suggest eight different Datasets for Modelling

DataSet	GRT high	Distance Threshold	Proximity Criterion	Deers	Observations
1	36	10	last	35	149
2	36	10	nearest	35	147
3	200	15	score	36	207

Agenda

The Background

The Data

The Models

The Wrap-up

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

Time difference [hours]
Distance [km]
Sample delay [hours]
Pregnant
Defecation day (between 1 and 366)
Number of other relevant hunting events

The Models

Model	Type	Non-Parametric Effects	Linear Effects	Random Intercept	Distribution Assumption
A	GAM	Time Difference, Distance, Sample Delay, Day of Year	Pregnant, Number Other Hunts	None	Gaussian
B	GAM	Time Difference, Distance, Sample Delay, Day of Year	Pregnant, Number Other Hunts	None	Gamma
C	GAMM	Time Difference, Distance, Sample Delay, Day of Year	Pregnant, Number Other Hunts	Deer	Gaussian
D	GAMM	Time Difference, Distance, Sample Delay, Day of Year	Pregnant, Number Other Hunts	Deer	Gamma

A Generalized Additive Model

i) Gaussian: Identity Link

\(FCM_i \sim \mathcal{N}(\mu_i, \sigma^2)\)
Identity Link: \(E(FCM_i) = \mu_i = \eta_i\)
Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

ii) Gamma: Log Link

\(FCM_i \sim \mathcal{Ga}(\nu, \frac{\nu}{\mu_i})\)
For better Interpretability we use the Log-Link: \(E(FCM_i) = \mu_i = exp(\eta_i)\)
Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{N}\left( \mu_{ij}, \sigma^2 \right) \\ \mu_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gamma: Log link

Log link for interpretability.
Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]